Gaz is the daughter of Professor Membrane, as well as his youngest child. It has been said, however, that Dib was created by the Professor in his lab and it was unknown for Gaz, and are therefore making Dib his non-biological child and Gaz unconfirmed. Nevertheless, Membrane still treats them as such... when he's around. Usually, the Professor is busy in his lab, and communicates with his children via recorded messages.
However, based on their live interactions, it is clear that the Professor and Gaz have a lot in common: Both of them have little patience for Dib's antics, for their own reasons (the Professor considers Dib insane, while Gaz just thinks he's annoying).
Professor Membrane[]
Professor Membrane often neglects both of his children, maintaining contact with them mostly through floating screens that are oftentimes pre-recorded. However, he shows affection towards them in limited occasions; in "Gaz, Taster of Pork", when the Professor can't invent a cure for his daughter's "disease", he becomes rather desperate. He "protects" Gaz by putting her in a special insulating suit and confining her to quarantine, then he builds an entire speciality area to try to find the cure.
They then go out to have dinner at Bloaty's after she is cured. In "Game Slave 2", he insists in her being escorted by Dib to the midnight Game Slave 2 video game console sale, because it could be too "dangerous" to go by herself, showing genuine worry. However, he does show willingness to let his very young children wander the streets at night, which leaves up to debate how much he really cares about their safety.
It can be argued that Gaz is Professor Membrane's favorite of the two, considering that her brother often makes a fool of himself with his paranormal preoccupation. He even goes so far as to state that Gaz should find Dib and "Make sure he doesn't do anything embarrassing." He shows a tendency of picking her up a lot, often to carry her on his shoulder. He has stated on occasion that "You're my funny child!" Compared to Dib, who would likely be his "insane child."
Gaz and the professor playing video games together in Issue 1
In several issues of the comic series, Membrane is shown to be a better father than in the TV series. He takes the time to play games with Gaz, showing a somewhat increased dedication towards spending time with her physically. He also eats several meals together with both of his children, though Foodio, their home robot, prepares the food when they eat at home. In Issue 14, when they're all eating at Bloaty's, he says that he's happy to be there, even if he finds the restaurant "questionable".
Gaz[]
Professor Membrane might be the only person to which Gaz shows a level of respect and love to, despite being neglected by him for most of her life except one day a year.
The biggest example of this is "Bloaty's Pizza Hog", showing that the "Family Dinner Night" (the only night in the year in which Gaz, Dib and Professor Membrane pass time together as a family) is important enough for her to go to save Dib, and bring him in time and unharmed.
It should be noticed that in this episode she doesn't care for Dib fighting with Zim in the night of the event, and tries to convince Professor Membrane to exclude Dib from the event, but when Professor Membrane refuses to go without Dib, it is then that she goes to save him.
She seems to be the only one of them that cares that they rarely get to see their father, and is therefore more willing to go out of her way to spend time with him when she can. Although she seems to admire her father, she seems to get extremely angered at him for lying to her in "Dibship Rising", perhaps the only seen time she ever does get angry at him.
In Invader Zim: Enter the Florpus, Gaz is upset on her father's behalf when she realizes Dib is planning to crash his keynote speech to expose Zim, and angry with him when this backfires and leaves the Professor imprisoned on Moo-Ping 10. When they manage to get Tak's Ship working and reach the prison, she's clearly happy to be reunited with him, as she's shown perching on his shoulder in a hug.
Several issues of the comic series show Gaz and Membrane playing video games together, but perhaps the best example in the comics comes from Quarterly 2. When she learns that Membrane's reputation has been damaged by his refusal to believe in the existence of the Beeboogoat (actually GIR covered in dead bees and garbage), and that he's become depressed because of it, she calls Dib and demands that he fix the situation, before traveling cross-country to join them. Arriving right after Dib has proven GIR isn't a Beeboogoat, she keeps him from complicating things by trying to expose Zim again, forcing him to stay focused on their father.